Bulg. J. Phys. 42 (2015) 236–248

Invariant Differential Operators forNon-Compact Lie Groups: an Introduction∗

V.K. DobrevInstitute for Nuclear Research and Nuclear Energy, Bulgarian Academy ofSciences, 72 Tsarigradsko Chaussee, 1784 Sofia, Bulgaria

Received 30 May 2015

Abstract. In the present paper we review the progress of the project of clas-sification and construction of invariant differential operators for non-compactsemisimple Lie groups. Our starting points is the class of algebras, which wecalled earlier ’conformal Lie algebras’ (CLA), which have very similar proper-ties to the conformal algebras of Minkowski space-time, though our aim is togo beyond this class in a natural way. For this we introduced recently the newnotion of parabolic relation between two non-compact semisimple Lie algebrasG and G′ that have the same complexification and possess maximal parabolicsubalgebras with the same complexification. In the present paper we consider indetail the orthogonal algebras so(p, q) all of which are parabolically related tothe conformal algebra so(n, 2) with p + q = n + 2, the parabolic subalgebrasincluding the Lorentz subalgebra so(n− 1, 1) and its analogs so(p− 1, q− 1).

PACS codes: 02.20.Tw

1 Introduction

Invariant differential operators play very important role in the description ofphysical symmetries – starting from the early occurrences in the Maxwell,d’Allembert, Dirac, equations, to the latest applications of (super-)differentialoperators in conformal field theory, supergravity and string theory (for reviews,cf. e.g., [1, 2]). Thus, it is important for the applications in physics to studysystematically such operators. For other relevant references cf., e.g., [3–6], andothers throughout the text.

In a recent paper [7] we started the systematic explicit construction of invariantdifferential operators. We gave an explicit description of the building blocks,namely, the parabolic subgroups and subalgebras from which the necessaryrepresentations are induced. Thus we have set the stage for study of differentnon-compact groups.∗Invited talk at Matey Mateev Symposium, Sofia, 17 April 2015.

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Invariant Differential Operators for Non-Compact Lie Groups: an Introduction

Since the study and description of detailed classification should be done groupby group we had to decide which groups to study. One first choice would benon-compact groups that have discrete series of representations. By the Harish-Chandra criterion [8] these are groups where holds

rankG = rankK,

where K is the maximal compact subgroup of the non-compact group G. An-other formulation is to say that the Lie algebra G of G has a compact Cartansubalgebra.

Example : the groups SO(p, q) have discrete series, except when both p, q areodd numbers.

This class is rather big, thus, we decided to consider a subclass, namely, the classof Hermitian symmetric spaces. The practical criterion is that in these cases, themaximal compact subalgebra K is of the form

K = so(2)⊕K′ . (1)

The Lie algebras from this class are

so(n, 2), sp(n,R), su(m,n), so∗(2n), E6(−14) , E7(−25) (2)

These groups/algebras have highest/lowest weight representations, and relatedlyholomorphic discrete series representations.

The most widely used of these algebras are the conformal algebras so(n, 2) inn-dimensional Minkowski space-time. In that case, there is a maximal Bruhatdecomposition [9] that has direct physical meaning

so(n, 2) =M ⊕ A ⊕ N ⊕ N ,

M = so(n− 1, 1) , dimA = 1, dimN = dim N = n ,(3)

where so(n − 1, 1) is the Lorentz algebra of n-dimensional Minkowski space-time, the subalgebra A = so(1, 1) represents the dilatations, the conjugatedsubalgebrasN , N are the algebras of translations, and special conformal trans-formations, both being isomorphic to n-dimensional Minkowski space-time.

The subalgebra P =M ⊕ A ⊕ N (∼=M ⊕ A ⊕ N ) is a maximal parabolicsubalgebra.

There are other special features which are important. In particular, the complexi-fication of the maximal compact subgroup is isomorphic to the complexificationof the first two factors of the Bruhat decomposition

KC = so(n,C)⊕ so(2,C) ∼= so(n− 1, 1)C ⊕ so(1, 1)C =MC ⊕AC . (4)

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In particular, the coincidence of the complexification of the semi-simple subal-gebras

K′C =MC (∗)means that the sets of finite-dimensional (nonunitary) representations ofM arein 1-to-1 correspondence with the finite-dimensional (unitary) representations ofK′. The latter leads to the fact that the corresponding induced representationsare representations of finite K-type [8].

It turns out that some of the hermitian-symmetric algebras share the above-mentioned special properties of so(n, 2). This subclass consists of:

so(n, 2), sp(n,R), su(n, n), so∗(4n), E7(−25) (5)

the corresponding analogs of Minkowski space-time V being

Rn−1,1, Sym(n,R), Herm(n,C), Herm(n,Q), Herm(3,O) . (6)

In view of applications to physics, we proposed to call these algebras conformalLie algebras, (or groups).

We have started the study of the above class in the framework of the presentapproach in the cases: so(n, 2) [10], su(n, n) [11], sp(n,R) [12], E7(−25) [13],so∗(12) [14], resp., and we have considered also the algebra E6(−14) [15].

Lately, we discovered an efficient way to extend our considerations beyond thisclass introducing the notion of ’parabolically related non-compact semisimpleLie algebras’ [16].

• Definition : Let G,G′ be two non-compact semisimple Lie algebras with thesame complexification GC ∼= G′C. We call them parabolically related if theyhave parabolic subalgebras P =M⊕A⊕N , P ′ =M′ ⊕A′ ⊕N ′, such that:MC ∼=M′C (⇒ PC ∼= P ′C). ♦Certainly, there are many such parabolic relationships for any given algebraG. Furthermore, two algebras G,G′ may be parabolically related via differentparabolic subalgebras.

We summarize the algebras parabolically related to conformal Lie algebraswith maximal parabolics fulfilling (∗) in Table 1, where we display only thesemisimple part K′ of K; sl(n,C)R denotes sl(n,C) as a real Lie alge-bra, (thus, (sl(n,C)R)C = sl(n,C) ⊕ sl(n,C)); e6 denotes the compact realform of E6 ; and we have imposed restrictions to avoid coincidences or degen-eracies due to well known isomorphisms: so(1, 2) ∼= sp(1,R) ∼= su(1, 1),so(2, 2) ∼= so(1, 2) ⊕ so(1, 2), su(2, 2) ∼= so(4, 2), sp(2,R) ∼= so(3, 2),so∗(4) ∼= so(3)⊕ so(2, 1), so∗(8) ∼= so(6, 2).

After this extended introduction we give the outline of the paper. In Section 2 wegive the preliminaries, actually recalling and adapting facts from [7]. In Section3 we consider the case of the pseudo-orthogonal algebras so(p, q) which areparabolically related to the conformal algebra so(n, 2) for p+ q = n+ 2 [17].

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Table 1. Conformal Lie algebras (CLA) G with M-factor fulfilling (∗) and the corre-sponding parabolically related algebras G′

G K′ M G′ M′

dimV

so(n, 2) so(n) so(n− 1, 1) so(p, q) so(p− 1, q − 1)n ≥ 3

n p+ q = n+ 2

su(n, n) su(n)⊕ su(n) sl(n,C)R sl(2n,R) sl(n,R)⊕ sl(n,R)n ≥ 3

n2 su∗(2n), n = 2k su∗(2k)⊕ su∗(2k)

sp(2r,R) su(2r) sl(2r,R) sp(r, r) su∗(2r)rank = 2r ≥ 4

r(2r + 1)

so∗(4n) su(2n) su∗(2n) so(2n, 2n) sl(2n,R)n ≥ 3

n(2n− 1)

E7(−25) e6 E6(−26) E7(7) E6(6)

27

2 Preliminaries

Let G be a semisimple non-compact Lie group, and K a maximal compact sub-group of G. Then we have an Iwasawa decomposition G = KA0N0, whereA0 is Abelian simply connected vector subgroup of G, N0 is a nilpotent sim-ply connected subgroup of G preserved by the action of A0. Further, let M0 bethe centralizer of A0 in K. Then the subgroup P0 = M0A0N0 is a minimalparabolic subgroup ofG. A parabolic subgroup P = M ′A′N ′ is any subgroupof G which contains a minimal parabolic subgroup.

Further, let G,K,P,M,A,N denote the Lie algebras of G,K,P,M,A,N ,resp.

For our purposes we need to restrict to maximal parabolic subgroups P =MAN , i.e. rankA = 1, resp. to maximal parabolic subalgebras P =M⊕A⊕N with dim A = 1.

Let ν be a (non-unitary) character ofA, ν ∈ A∗, parameterized by a real numberd, called the conformal weight or energy.

Further, let µ fix a discrete series representation Dµ of M on the Hilbert spaceVµ, or the finite-dimensional (non-unitary) representation of M with the sameCasimirs.

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We call the induced representation χ = IndGP (µ⊗ν⊗1) an elementary represen-tation of G [18]. (These are called generalized principal series representations(or limits thereof) in [19].) Their spaces of functions are

Cχ = {F ∈ C∞(G,Vµ)|F(gman) = e−ν(H) ·Dµ(m−1)F(g)}, (7)

where a = exp(H) ∈ A′, H ∈ A′ , m ∈ M ′, n ∈ N ′. The representationaction is the left regular action

(T χ(g)F)(g′) = F(g−1g′), g, g′ ∈ G . (8)

• An important ingredient in our considerations are the highest/lowest weightrepresentations of GC. These can be realized as (factor-modules of) Verma mod-ules V Λ over GC, where Λ ∈ (HC)∗, HC is a Cartan subalgebra of GC, weightΛ = Λ(χ) is determined uniquely from χ [20].

Actually, since our ERs may be induced from finite-dimensional representationsof M (or their limits) the Verma modules are always reducible. Thus, it ismore convenient to use generalized Verma modules V Λ such that the role ofthe highest/lowest weight vector v0 is taken by the (finite-dimensional) spaceVµ v0 . For the generalized Verma modules (GVMs) the reducibility is controlledonly by the value of the conformal weight d. Relatedly, for the intertwiningdifferential operators only the reducibility w.r.t. non-compact roots is essential.

• Another main ingredient of our approach is as follows. We group the (re-ducible) ERs with the same Casimirs in sets called multiplets [20]. The multipletcorresponding to fixed values of the Casimirs may be depicted as a connectedgraph, the vertices of which correspond to the reducible ERs and the lines (ar-rows) between the vertices correspond to intertwining operators. The explicitparametrization of the multiplets and of their ERs is important for understand-ing of the situation. The notion of multiplets was introduced in [21, 22] andapplied to representations of SOo(p, q) and SU(2, 2), resp., induced from theirminimal parabolic subalgebras. Then it was applied to (infinite-dimensional)(super-)algebras, quantum groups and other symmetry objects.

In fact, the multiplets contain explicitly all the data necessary to construct theintertwining differential operators. Actually, the data for each intertwining dif-ferential operator consists of the pair (β,m), where β is a (non-compact) posi-tive root of GC, m ∈ N, such that the BGG Verma module reducibility condition(for highest weight modules) is fulfilled

(Λ + ρ, β∨) = m , β∨ ≡ 2β/(β, β) . (9)

ρ is half the sum of the positive roots of GC. When the above holds then theVerma module with shifted weight V Λ−mβ (or V Λ−mβ for GVM and β non-compact) is embedded in the Verma module V Λ (or V Λ). This embedding is

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realized by a singular vector vs determined by a polynomial Pm,β(G−) in theuniversal enveloping algebra (U(G−)) v0 , G− is the subalgebra of GC generatedby the negative root generators [23]. More explicitly, [20], vsm,β = Pm,β v0 (orvsm,β = Pm,β Vµ v0 for GVMs). Then there exists [20] an intertwining differen-tial operator

Dm,β : Cχ(Λ) −→ Cχ(Λ−mβ) (10)

given explicitly byDm,β = Pm,β(G−) , (11)

where G− denotes the right action on the functions F .

In most of these situations the invariant operatorDm,β has a non-trivial invariantkernel in which a subrepresentation of G is realized. Thus, studying the equa-tions with trivial RHS

Dm,β f = 0 , f ∈ Cχ(Λ) (12)

is also very important. For example, in many physical applications in the caseof first order differential operators, i.e., for m = mβ = 1, these equations arecalled conservation laws, and the elements f ∈ kerDm,β are called conservedcurrents.

The above construction works also for the subsingular vectors vssv of Vermamodules. Such a vector is also expressed by a polynomial Pssv(G−) in theuniversal enveloping algebra vsssv = Pssv(G−) v0 , cf. [24]. Thus, there ex-ists a conditionally invariant differential operator given explicitly by Dssv =

Pssv(G−), and a conditionally invariant differential equation, for many moredetails, see [24]. (Note that these operators (equations) are not of first order.)

Below in our exposition we shall use the so-called Dynkin labels

mi ≡ (Λ + ρ, α∨i ) , i = 1, . . . , n , (13)

where Λ = Λ(χ), ρ is half the sum of the positive roots of GC.

We shall use also the so-called Harish-Chandra parameters

mβ ≡ (Λ + ρ, β) , (14)

where β is any positive root of GC. These parameters are redundant, since theyare expressed in terms of the Dynkin labels, however, some statements are bestformulated in their terms. (Clearly, both the Dynkin labels and Harish-Chandraparameters have their origin in the BGG reducibility condition (9).)

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3 Conformal Algebras so(n, 2) and Parabolically Related

Let G = so(n, 2), n > 2. We label the signature of the ERs of G as follows:

χ = {n1, . . . , nh ; c}, nj ∈ Z/2 , c = d− n2 , h ≡ [n2 ]

|n1| < n2 < · · · < nh , n even ,

0 < n1 < n2 < · · · < nh , n odd ,

(15)

where the last entry of χ labels the characters of A , and the first h entries arelabels of the finite-dimensional nonunitary irreps ofM∼= so(n− 1, 1).

The reason to use the parameter c instead of d is that the parametrization of theERs in the multiplets is given in a simple intuitive way (cf. [10, 25])

χ±1 = {εn1 , . . . , nh ; ±nh+1} , nh < nh+1 ,

χ±2 = {εn1 , . . . , nh−1 , nh+1 ; ±nh}χ±3 = {εn1,. . . ,nh−2,nh,nh+1 ; ±nh−1}

· · · · · · · · · · · · · · · · · · · · · · · · · · ·χ±h

= {εn1 , n3 , . . . , nh , nh+1 ; ±n2}χ±h+1

= {εn2 , . . . , nh , nh+1 ; ±n1}

ε =

{± , n even1, n odd

(16)

Figure 1. Simplest example of diagram withconformal invariant operators (arrows are dif-ferential operators, dashed arrows are integraloperators).

∂µ =∂

∂µ,

Aµ – electromagnetic potential,

∂µφ = Aµ ,

F – electromagnetic field,

∂[λ,Aµ] = ∂λAµ − ∂µAλ = Fλµ ,

Jµ – electromagnetic current,

∂µFλµ = Jµ, ∂µJµ = Φ.

�� / /

�� / /

φ Φ

Aμ Jμ

F[λ,μ]

∂μ ∂ μ

∂[λ,·] ∂λ

�

�

������������������

Fig. 1. Simplest example of diagram with conformal invariant operators

(arrows are differential operators, dashed arrows are integral operators)

∂μ = ∂∂ xµ

, Aμ electromagnetic potential, ∂μ φ = Aμ

F electromagnetic field, ∂[λAμ] = ∂λAμ − ∂μAλ = Fλμ

Jμ electromagnetic current, ∂λFλμ = Jμ, ∂μJμ = Φ

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Invariant Differential Operators for Non-Compact Lie Groups: an Introduction

Further, we denote by C±i the representation space with signature χ±i .

The number of ERs in the corresponding multiplets is equal to

|W (GC,HC)| / |W (MC,HCm)| = 2(1 + h) , (17)

where HC, HCm are Cartan subalgebras of GC, MC, resp. This formula is valid

for the main multiplets of all conformal Lie algebras.

We show some examples of diagrams of invariant differential operators for theconformal groups so(5, 1), resp. so(4, 2), in 4-dimensional Euclidean, resp.Minkowski, space-time. In Figure 1 we show the simplest example for the mostcommon using well known operators. In Figure 2 we show the same examplebut using the group-theoretical parity splitting of the electromagnetic current,cf. [26]. In Figure 3 we show the general classification for so(5, 1) given in [26].These diagrams are valid also for so(4, 2) [27] and for so(3, 3) ∼= sl(4,R) [16].

�� / /

�� / /

/ / ��

φ Φ

Aμ Jμ

F−[λ,μ] F+

[λ,μ]

∂μ ∂ μ

d2 d2d3d3

�

�

�

�������������������

���������������

Fig. 2. More precise showing of the simplest example,

F = F+ ⊕ F− shows the parity splitting of the electromagnetic field,

d2, d3 linear invariant operators

Figure 2. More precise showing of the sim-plest example, F = F+ ⊕ F− shows theparity splitting of the electromagnetic field,d2, d3 – linear invariant operators.

�� / /

�� / /

/ / ��

Λ−pνn Λ+

pνn

Λ′−pνn Λ′+

pνn

Λ′′−pνn Λ′′+

pνn

dν1 d′ν1

dn2 dn2dp3dp3

�

�

�

���

��

��

��

��

��

��

�����

��

��

��

��

��

��

��

Fig. 3. The general classification of invariant differential operators valid for

so(4, 2), so(5, 1) and so(3, 3) ∼= sl(4,R).p, ν, n are three natural numbers, the shown simplest case is when p = ν = n = 1,

dν1 is a linear differential operator of order ν, similarly d′ν1 , dn2 , dp3

Figure 3. The general classificationof invariant differential operators validfor so(4, 2), so(5, 1), and so(3, 3) ∼=sl(4,R). p, ν, n are three natural numbers,the shown simplest case is when p = ν =n = 1, dν1 is a linear differential operatorof order ν, similarly d′ν1 , dn2 , dp3 .

243

V.K. Dobrev

Next in Figure 4 we show the general even case so(p, q), p+q = 2h+2, [10,25],while in Figure 5. we show an alternative view of the same case.

Next in Figure 6 we show the general odd case so(p, q), p+q = 2h+3, [10,25],while in Figure 7 we show an alternative view of the same case.

��

��

��

��

� � � � � � � � � � � � � �

Λ−1 Λ+

1/ /

Λ−2 Λ+

2/ /

Λ−h Λ+

h/ /

/ /Λ−h+1 Λ+

h+1

d1 d′1

d2 d′2

dh−1 d′h−1

dh dhdh+1dh+1

�

�

�

�

�

�

�

�������������������

����������������

Fig. 4. The general classification of invariant differential operators in 2h-dimensional space-time.

By parabolic relation the diagram above is valid for all algebras so(p, q), p+ q = 2h+ 2 , even.

Figure 4. The general classificationof invariant differential operators in 2h-dimensional space-time. By parabolic re-lation the diagram above is valid for all al-gebras so(p, q), p+ q = 2h+ 2, even.

Λ−1

Λ−2

d1

...

Λ−h−1

�

�

dh−1

Λ−h������

������

dh dh+1Λ−

h+1 Λ+h+1• ������

������dh+1 dh

d′h−1

d′1

Λ+h

�

�

Λ+h−1

...

Λ+2

Λ+1

Fig. 5. Alternative showing of the case so(p, q), p+ q = 2h+ 2, showing only the differentialoperators, while the integral operators are assumed as symmetry w.r.t. the bullet in the centre.

Figure 5. Alternative showing of the caseso(p, q), p + q = 2h + 2, showing onlythe differential operators, while the integraloperators are assumed as symmetry w.r.t.the bullet in the centre.

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Invariant Differential Operators for Non-Compact Lie Groups: an Introduction

��

��

��

��

� � � � � � � � � � � � � �

/ /Λ−1 Λ+

1

/ /Λ−2 Λ+

2

/ /Λ−h Λ+

h

/Λ−h+1 Λ+

h+1

dh+1

d1 d′1

d2 d′2

dh−1 d′h−1

dh d′h

�

�

�

�

�

�

�

�

Fig. 6. The general classification of invariant differential operators in 2h+ 1 dimensional space-time.

By parabolic relation the diagram above is valid for all algebras so(p, q), p+ q = 2h+ 3 , odd.

Figure 6. The general classification of in-variant differential operators in 2h + 1 di-mensional space-time. By parabolic rela-tion the diagram above is valid for all alge-bras so(p, q), p+ q = 2h+ 3, odd.

Λ−1

Λ−2

d1

...

Λ−h

�

�

dh

dh+1

Λ−h+1

�

•

d′h

d′1

Λ+h+1

�

�

Λ+h

...

Λ+2

Λ+1

Fig. 7. Alternative showing of the case so(p, q), p+ q = 2h+ 3, showing only the differentialoperators, while the integral operators are assumed as symmetry w.r.t. the bullet in the centre.

Figure 7. Alternative showing of the caseso(p, q), p + q = 2h + 3, showing onlythe differential operators, while the integraloperators are assumed as symmetry w.r.t.the bullet in the centre.

The ERs in the multiplet are related by intertwining integral and differentialoperators. The integral operators were introduced by Knapp and Stein [28].They correspond to elements of the restricted Weyl group of G. These operatorsintertwine the pairs C±i

G±i : C∓i −→ C±i , i = 1, . . . , 1 + h (18)

The intertwining differential operators correspond to non-compact positive rootsof the root system of so(n+2,C), cf. [20]. [In the current context, compact roots

245

V.K. Dobrev

of so(n+ 2,C) are those that are roots also of the subalgebra so(n,C), the restof the roots are non-compact.] The degrees of these intertwining differentialoperators are given just by the differences of the c entries [25]

deg di = deg d′i = nh+2−i − nh+1−i , i = 1, . . . , h , ∀ndeg dh+1 = n2 + n1 , n even

(19)

where d′h is omitted from the first line for (p+ q) even.

Matters are arranged so that in every multiplet only the ER with signature χ−1contains a finite-dimensional nonunitary subrepresentation in a subspace E . Thelatter corresponds to the finite-dimensional unitary irrep of so(n+2) with signa-ture {n1 , . . . , nh , nh+1}. The subspace E is annihilated by the operator G+

1 ,and is the image of the operator G−1 .

Although the diagrams are valid for arbitrary so(p, q) (p + q ≥ 5) the contentsis very different. We comment only on the ER with signature χ+

1 . In all casesit contains an UIR of so(p, q) realized on an invariant subspace D of the ERχ+

1 . That subspace is annihilated by the operator G−1 , and is the image of theoperator G+

1 . (Other ERs contain more UIRs.)

If pq ∈ 2N the mentioned UIR is a discrete series representation. (Other ERscontain more discrete series UIRs.)

And if q = 2 the invariant subspace D is the direct sum of two subspacesD = D+ ⊕ D−, in which are realized a holomorphic discrete series represen-tation and its conjugate anti-holomorphic discrete series representation, resp.Note that the corresponding lowest weight GVM is infinitesimally equivalentonly to the holomorphic discrete series, while the conjugate highest weight GVMis infinitesimally equivalent to the anti-holomorphic discrete series.

Note that the deg di , deg d′i , are Harish-Chandra parameters corresponding tothe non-compact positive roots of so(n+ 2,C). From these, only deg d1 corre-sponds to a simple root, i.e., is a Dynkin label.

Above we considered so(n, 2) for n > 2. The case n = 2 is reduced to n = 1since so(2, 2) ∼= so(1, 2) ⊕ so(1, 2). The case so(1, 2) is special and must betreated separately. But in fact, it is contained in what we presented already. Inthat case the multiplets contain only two ERs which may be depicted by the toppair χ±1 in the pictures that we presented. And they have the properties thatwe described for so(n, 2) with n > 2. The case so(1, 2) was given already in1946-7 independently by Gel’fand et al [29] and Bargmann [30].

Acknowledgments

The author has received partial support from Bulgarian NSF Grant DFNI T02/6and from COST action MP-1210.

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Invariant Differential Operators for Non-Compact Lie Groups: an Introduction

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[19] A.W. Knapp (1986) Representation Theory of Semisimple Groups (An OverviewBased on Examples), Princeton Univ. Press.

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and E.M. Stein (1980) Inventiones Math. 60 9-84.[29] I.M. Gelfand and M.A. Naimark (1946) Acad. Sci. USSR. J. Phys. 10 93-94.[30] V. Bargmann (1947), Irreducible unitary representations of the Lorentz group, An-

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